Any solution of this form is rotating with a time independent angular velocity \(\Omega \in \mathbb {R}\)  around its center of mass fixed at the origine. If \(\Omega =0,\)  then it is stationary, otherwise it is time periodic with period \(\tfrac{2\pi }{\Omega }.\)  Historically, the first example of V-states was provided by Kirchhoff [1]} who showed that for Euler equations any ellipse with semi-axis \(a\)  and \(b\)  uniformly rotates for an angular velocity \(\Omega =\tfrac{ab}{(a+b)^2}.\)  Observe from (REF ), (REF ) and (REF ) that the stream function \(\Psi \)  admits a radial Green kernel. Therefore, any radial initial profile would generate a stationary solution. In particular, the Rankine vortices given by the discs provide such examples. The purpose of this study is to prove the existence of periodic (and quasi-periodic) vortex patch structures close to these equilibrium states. Due to the invariance by dilation of the model, it is sufficient to search for these type of solutions close to the unit disc. The analytical study of existence of V-states close to the unit disc for Euler equations goes back to the work of Burbea [2]}. We also refer to [3]} for a more recent and rigorous point of view. Indeed, combining Crandall-Rabinowitz's Theorem REF  and complex analysis, one can find the existence of branches of \(\mathbf {m}\) -fold V-states bifurcating from the unit disc at the angular velocities
\(\Omega _{\mathbf {m}}^{\textnormal {\tiny {E}}}\triangleq \frac{\mathbf {m}-1}{2\mathbf {m}}.\) 
